Semilinear transformation

In linear algebra, particularly projective geometry, a semilinear transformation between vector spaces V and W over a field K is a function that is a linear transformation "up to a twist", hence semi-linear, where "twist" means "field automorphism of K". Explicitly, it is a function $T\colon V \to W$ that is:

• linear with respect to vector addition: $T(v+v') = T(v)+T(v')$
• semilinear with respect to scalar multiplication: $T(\lambda v) = \lambda^\theta T(v),$ where θ is a field automorphism of K, and $\lambda^\theta$ means the image of the scalar $\lambda$ under the automorphism. There must be a single automorphism θ for T, in which case T is called θ-semilinear.

The invertible semilinear transforms of a given vector space V (for all choices of field automorphism) form a group, called the general semilinear group and denoted $\operatorname{\Gamma L}(V),$ by analogy with and extending the general linear group.

Similar notation (replacing Latin characters with Greek) are used for semilinear analogs of more restricted linear transform; formally, the semidirect product of a linear group with the Galois group of field automorphism. For example, PΣU is used for the semilinear analogs of the projective special unitary group PSU. Note however, that it is only recently noticed that these generalized semilinear groups are not well-defined, as pointed out in (Bray, Holt & Roney-Dougal 2009) – isomorphic classical groups G and H (subgroups of SL) may have non-isomorphic semilinear extensions. At the level of semidirect products, this corresponds to different actions of the Galois group on a given abstract group, a semidirect product depending on two groups and an action. If the extension is non-unique, there are exactly two semilinear extensions; for example, symplectic groups have a unique semilinear extension, while SU(n,q) has two extension if n is even and q is odd, and likewise for PSU.

Definition

Let K be a field and k its prime subfield. For example, if K is C then k is Q, and if K is the finite field of order $q=p^i,$ then k is $\mathbf{Z}/p\mathbf{Z}.$

Given a field automorphism $\theta$ of K, a function $f\colon V \to W$ between two K vector spaces V and W is $\theta$-semilinear, or simply semilinear, if for all $x,y$ in V and $l$ in K it follows:

1. $f(x+y)=f(x)+f(y),$
2. $f(lx)=l^\theta f(x),$

where $l^\theta$ denotes the image of $l$ under $\theta.$

Note that $\theta$ must be a field automorphism for f to remain additive, for example, $\theta$ must fix the prime subfield as

$n^\theta f(x)=f(nx)=f(x+\dots +x)=nf(x)$

Also

$(l_1+l_2)^\theta f(x)=f((l_1+l_2)x)=f(l_1 x)+f(l_2 x)=(l_1^\theta+l_2^\theta)f(x)$

so $(l_1+l_2)^\theta=l_1^\theta+l_2^\theta.$ Finally,

$(l_1 l_2)^\theta f(x)=f(l_1 l_2 x)=l_1^\theta f(l_2x)=l_1^\theta l_2^\theta f(x)$

Every linear transformation is semilinear, but the converse is generally not true. If we treat V and W as vector spaces over k, (by considering K as vector space over k first) then every $\theta$-semilinear map is a k-linear map, where k is the prime subfield of K.

Examples

• Let $K=\mathbf{C}, V=\mathbf{C}^n,$ with standard basis $e_1,\ldots, e_n.$ Define the map $f\colon V \to V$ by

$f\left(\sum_{i=1}^n z_i e_i \right) = \sum_{i=1}^n \bar z_i e_i$

f is semilinear (with respect to the complex conjugation field automorphism) but not linear.

• Let $K=GF(q)$ – the Galois field of order $q=p^i,$ p the characteristic. Let $l^\theta = l^p.$ By the Freshman's dream it is known that this is a field automorphism. To every linear map $f\colon V \to W$ between vector spaces V and W over K we can establish a $\theta$-semilinear map

$\widetilde{f} \left( \sum_{i=1}^n l_i e_i\right) := f \left( \sum_{i=1}^n l_i^\theta e_i \right)$

Indeed every linear map can be converted into a semilinear map in such a way. This is part of a general observation collected into the following result.

General semilinear group

Given a vector space V, the set of all invertible semilinear maps (over all field automorphisms) is the group $\operatorname{\Gamma L}(V).$

Given a vector space V over K, and k the prime subfield of K, then $\operatorname{\Gamma L}(V)$ decomposes as the semidirect product

$\operatorname{\Gamma L}(V) = \operatorname{GL}(V) \rtimes \operatorname{Aut}(K/k)$

where Aut(K/k) is the automorphisms of $K$ as a field over $k$. Similarly, semilinear transforms of other linear groups can be defined as the semidirect product with the automorphism group, or more intrinsically as the group of semilinear maps of a vector space preserving some properties.

We identify $\operatorname{Aut}(K/k)$ with a subgroup of $\operatorname{\Gamma L}(V)$ by fixing a basis B for V and defining the semilinear maps:

$\sum_{b\in B} l_b b \mapsto \sum_{b \in B} l_b^\sigma b$

for any $\sigma \in \operatorname{Aut}(K/k).$ We shall denoted this subgroup by Aut(K/k)_B. We also see these complements to GL(V) in $\operatorname{\Gamma L}(V)$ are acted on regularly by GL(V) as they correspond to a change of basis.

Proof

Every linear map is semilinear, thus $\operatorname{GL}(V) \leq \operatorname{\Gamma L}(V).$ Fix a basis B of V. Now given any semilinear map f with respect to a field automorphism $\sigma \in \operatorname{Aut}(K/k),$ then define $g\colon V \to V$ by

$g \left(\sum_{b \in B} l_b b\right) := \sum_{b \in B}f \left(l_b^{\sigma^{-1}} b\right) = \sum_{b \in B} l_b f (b)$

As f(B) is also a basis of V, it follows that g is simply a basis exchange of V and so linear and invertible: $g \in \operatorname{GL}(V).$

Set $h:=f g^{-1}.$ For every $v=\sum_{b \in B} l_b b$ in V,

$hv=fg^{-1}v=\sum_{b \in B} l_b^\sigma b$

thus h is in the Aut(K/k) subgroup relative to the fixed basis B. This factorization is unique to the fixed basis B. Furthermore, GL(V) is normalized by the action of Aut(K/k)_B, so $\operatorname{\Gamma L}(V) = \operatorname{GL}(V) \rtimes \operatorname{Aut}(K/k).$

Applications

Projective geometry

The $\operatorname{\Gamma L}(V)$ groups extend the typical classical groups in GL(V). The importance in considering such maps follows from the consideration of projective geometry. The induced action of $\operatorname{\Gamma L}(V)$ on the associated vector space P(V) yields the projective semilinear group, denoted $\operatorname{P\Gamma L}(V),$ extending the projective linear group, PGL(V).

The projective geometry of a vector space V, denoted PG(V), is the lattice of all subspaces of V. Although the typical semilinear map is not a linear map, it does follow that every semilinear map $f\colon V \to W$ induces an order-preserving map $f\colon PG(V) \to PG(W).$ That is, every semilinear map induces a projectivity. The converse of this observation (except for the projective line) is the fundamental theorem of projective geometry. Thus semilinear maps are useful because they define the automorphism group of the projective geometry of a vector space.

Mathieu group

The group PΓL(3,4) can be used to construct the Mathieu group M₂₄, which is one of the sporadic simple groups; PΓL(3,4) is a maximal subgroup of M₂₄, and there are many ways to extend it to the full Mathieu group.

References

• Gruenberg, K. W. and Weir, A.J. Linear Geometry 2nd Ed. (English) Graduate Texts in Mathematics. 49. New York – Heidelberg – Berlin: Springer-Verlag. X, 198 pp. (1977).
• Bray, John N.; Holt, Derek F.; Roney-Dougal, Colva M. (2009), "Certain classical groups are not well-defined", Journal of Group Theory 12 (2): 171–180, doi:10.1515/jgt.2008.069, ISSN 1433-5883, MR 2502211

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